Inflation in heterotic supergravity models with torsion

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1 Inflation in heterotic supergravity models with torsion Stephen Angus IBS-CTPU, Daejeon in collaboration with Cyril Matti (City Univ., London) and Eirik Eik Svanes (LPTHE, Paris) (work in progress) String Phenomenology 2017 Virginia Tech, USA Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

2 Outline 1 Background Inflation: an overview Heterotic strings 2 Heterotic models with torsion SU(3) structure and half-flat manifolds Moduli stabilization with aligned D-terms 3 Inflation in heterotic models with torsion Simple case: one non-perturbative term Extension: two-axion model 4 Perturbative scenarios Inflation along the t and τ directions Consistency conditions Dilaton inflation Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

3 Background Inflation: an overview Why inflation? Some problems in Cosmology: Why is the universe spatially flat? How did the Cosmic Microwave Background (CMB) become isotropic at scales beyond the particle horizon? Proposed solution: Inflation. Universe experienced a period of exponentially rapid expansion. The comoving Hubble radius (ah) 1 would have been small and decreasing during inflation flat universe today. The horizon would have also been decreasing, so distant patches of the sky could have been in causal contact before inflation. Problem: to trust any model of inflation (quantum corrections...), should embed it in a UV-complete theory, eg. string theory. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

4 Background Heterotic strings Moduli stabilization String compactification gives rise to scalar fields called moduli. Types of moduli include: The (axio-)dilaton S sets the string coupling g s ; Kähler moduli T i size of 2-cycles in the compact geometry; Complex structure moduli Z a 3-cycles, shape of geometry. Moduli are flat directions in the potential this is catastrophic! Moduli need masses to prevent decompactification, 5th forces, etc. problem of moduli stabilization. Example: type IIB string theory: Compactify such that on the manifold, R-R 3-form flux F 3 and NS-NS flux H 3 are non-zero flux compactification. Kähler moduli remain unstabilized, can fix with eg. non-perturbative effects (KKLT), perturbative α corrections (LVS). All moduli stabilized! Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

5 Background Heterotic strings Challenges for heterotic moduli stabilization In heterotic string theory, only have NS-NS flux H 3. Can stabilize complex structure moduli... then what? Dilaton can be stabilized by gaugino condensation. Limited options for remaining moduli (worldsheet instantons...) In fact, problem is even worse: Strominger, 1986 If a heterotic compactification on a manifold Y has a maximally symmetric vacuum and non-vanishing H 3, Y is non-calabi Yau. Hence for a Calabi Yau compactification, H 3 = 0! Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

6 Heterotic models with torsion SU(3) structure and half-flat manifolds What is an SU(3) structure manifold? Mirror dual of H 3 : manifold with SU(3) structure hep-th/ (Vafa), hep-th/ (Gurrieri et al). SU(3) structure: there is a globally-defined spinor ζ that leaves 1/4 of the SUSY unbroken. Calabi Yau case: ζ is covariantly constant w.r.t. Levi-Civita. Non-CY case: ζ T 0 ζ (T 0 is intrinsic torsion of the manifold). Loophole in Strominger s theorem: Study compactifications on SU(3) structure manifolds with torsion. hep-th/ (Gurrieri et al), hep-th/ (de Carlos et al). Torsion quantization understood for half-flat manifolds these allow a superpotential linear in the Kähler moduli T i. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

7 Heterotic models with torsion Moduli stabilization with aligned D-terms Moduli stabilization with aligned D-terms: a review Consider a toy model: a half-flat manifold with axio-dilaton S = s + iσ and two Kähler moduli, T = t + iτ and U = u + iν. Assume complex structure moduli absent, or stabilized already. Allow geometric flux on the 2-cycle t only, giving a perturbative superpotential W P = w + e 1 T. Here w is generated by either NS flux or α corrections. Assume an anomalous U(1) symmetry under which only S and U transform, giving D-terms of the form D = b/s + c/u (b, c real). Resulting scalar potential stabilizes all moduli; minimum is non-supersymmetric AdS at (Lukas, Lalak, Svanes) s = 4bc e 2 1, t = w e 1, u = 4c2 3e 2 1, τ = 0. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

8 Inflation in heterotic models with torsion Simple case: one non-perturbative term Inflation: simple case A successful model of inflation must: - allow a de Sitter phase, V > 0; - satisfy the slow-roll conditions, ɛ, η 1. Need to identify a flat direction for the inflaton field. Promising candidates: two axions (σ and ν) remain unstabilized. One linear combination is gauge-dependent; it will be absorbed by the U(1) gauge boson via the Stueckelberg mechanism. Lift the remaining gauge-invariant axionic direction with a non-perturbative term eg (Ali, Haque, Jejjala), W NP = Ae α(s βu), which can arise from gaugino condensation in the hidden sector. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

9 Inflation in heterotic models with torsion Simple case: one non-perturbative term We expect such models to have a Kähler potential of the form K = ln s ln κ, κ = κ ijk t i t j t k where d ijk are intersection numbers on the mirror manifold. In the particular model above, we choose κ = t 2 u in order to stabilize all moduli perturbatively. The final vacuum is non-supersymmetric AdS, so need to uplift: assume this can be done such that V final = 0. Resulting potential has a natural inflation form ( )) ˆθ V = V 0 (1 cos, f where ˆθ is canonically normalized and the axion decay constant f = 1 α 2(1 + β 2 ). Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

10 Inflation in heterotic models with torsion Extension: two-axion model Two-axion model? Natural inflation with a single axion field requires a trans-planckian field excursion to match observations. Consider adding an additional Kähler modulus, X = x + iξ. Stabilize ξ with a worldsheet instanton superpotential, W NP2 = Be n 1U n 2 X, which is gauge-invariant by construction. Is moduli stabilization possible? Unclear (work in progress). For n 1 n 2 and β 1, the axions are almost aligned along the ν direction variation of aligned natural inflation? hep-ph/ (Kim, Nilles, Peloso) However, this violates the Weak Gravity Conjecture. (See talks by Shiu, Heidenreich, Rudelius, McAllister, Valenzuela, Blumenhagen, Hebecker, Ooguri,... ) Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

11 Perturbative scenarios A more elegant approach? A big advantage of moduli stabilization with aligned D-terms is that all moduli are stabilized perturbatively. Therefore we retain greater control if we can achieve inflation using only perturbative ingredients. One potential issue: typically not possible to stabilize all axions using only the perturbative potential. During inflation, any displacement in unstabilized axion directions can generate isocurvature perturbations, which are not observed in nature (Planck collaboration) However these problematic perturbations can be diluted if the inflation scale is sufficiently higher than the BBN scale. Also, difficult to realize Minkowski/de Sitter while stabilizing all moduli include uplifting term by hand (unsatisfying...). Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

12 Perturbative scenarios Choosing an inflaton candidate We would like to choose one of the moduli as a candidate inflaton. Expanding around the minimum gives the Hessian, H = k cr b r, 9br 0 0 r c where the order of modes is (τ, δt, δs, δu), and we have defined k = 3e8 1 8bc 3 w 2, r = 3e2 1 w 2 64bc 3. Since τ is an axion, it is protected by a discrete shift symmetry against quantum corrections, making it a good inflaton candidate. Note that τ is one of the light modes if r 1 (but so is t)! Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

13 Perturbative scenarios Inflation using t and τ Inflation along the t and τ directions Integrating out s and u by setting them to their VEVs, the uplifted potential for t and τ is ( ) V total = 3e6 1 16bc 3 1 2w e 1 t + w 2 + e1 2τ 2 (e 1 t) 2. For example, inflation mostly along the t-direction gives a Starobinsky-like potential eg (Blumenhagen et al). ( V τ=0 = 3e6 1 16bc 3 1 w ) 2 e ϕ, e 1 where ϕ = ln t is the canonically normalized inflaton field. For e-folds, field excursion ϕ ϕ CMB ϕ end 3.8 M P. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

14 Perturbative scenarios Inflation along the t and τ directions Inflation along the t-direction Figure: The t and τ-dependent potential, for k = w = e 1 = 1. This shows V (t, τ) (left) and the flow of its gradient (right). For O(1) parameters, the line τ = 0 is an attractor solution. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

15 Perturbative scenarios Inflation along the τ-direction Inflation along the t and τ directions Alternatively, consider inflation along the τ-direction. The potential at t = w/e 1 is V t= w e 1 = 3e bc 3 w 2 τ 2. This has a standard m 2 φ 2 form, with slow-roll parameters ɛ τ (τ) = η τ (τ) = 2 τ 2. For e-folds of inflation we require τ 15. But is this stable? We should also consider the height of the potential barrier in the t-direction. At large t, the potential V (t, τ) becomes V t = 3e6 1 16bc 3. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

16 Perturbative scenarios Inflation along the t and τ directions Figure: The t and τ-dependent potential, for w = 20 and k = e 1 = 1. A necessary condition for stability is that V t=w/e (τ) < V t across the entire trajectory of τ, meaning w/e 1 τ CMB 15. However, increasing w/e 1 creates a shallower barrier in the t-direction, so additional field excursion along t is expected. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

17 Perturbative scenarios Constraints on parameters Consistency conditions In heterotic models the strong coupling expansion should hold at the minimum: s t and s u 4bc we 1, b c. Meanwhile, the mass constraint for light t and τ, r 1, implies ( ) bc we Combining the constraints and the supergravity approximation yields a hierarchy of parameters, b, w c e 1. In the limit b c, the mass eigenstates of s and u become m+ 2 81e bc 7, m2 e b 3 c 5, with eigenvectors δx + δu and δx δs mass hierarchy, m 2 + m 2 τ m 2 t m 2. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

18 Perturbative scenarios Inflation in the s-direction Dilaton inflation V(s) s/(bc/e12 ) Inflation along s: Starobinsky-like model (ϕ ϕ 0 ln(s)/ 2), [ V = e6 0 8bc e 2ϕ e 2 2ϕ + 1 ] 8 e 3 2ϕ. Spectral index and tensor-to-scalar ratio consistent with data, 1.0 n s 0.965, r However, e-folds requires ϕ 2.9 M P likely unstable. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19

19 Summary Summary Various possibilities for inflation in heterotic models with torsion. Using aligned D-terms all moduli can be stabilized, leaving only massless axions. Resulting vacuum is non-supersymmetric AdS. Including non-perturbative contributions from gaugino condensation and worldsheet instantons natural inflation eg (Abe, Kobayashi, Otsuka). With two light axions, aligned natural inflation may be possible. Using perturbative terms only, scenarios appear such as m 2 φ 2 axion monodromy and Starobinsky-like models. However: - Is Minkowski/de Sitter possible while stabilizing all moduli? - in concrete scenarios, the light modes relevant for inflation are determined from constraints/parameter hierarchies. - ϕ > M P incompatible with quantum corrections/wgc. Stephen Angus Inflation in heterotic models with torsion 07/06/ / 19